4.14.49 \(x \left (x^3-2 y(x)^3\right ) y'(x)=y(x) \left (2 x^3-y(x)^3\right )\)

ODE
\[ x \left (x^3-2 y(x)^3\right ) y'(x)=y(x) \left (2 x^3-y(x)^3\right ) \] ODE Classification

[[_homogeneous, `class A`], _rational, _dAlembert]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.106763 (sec), leaf count = 331

\[\left \{\left \{y(x)\to \frac {\sqrt [3]{2} \left (\sqrt {81 x^6-12 e^{3 c_1} x^3}-9 x^3\right ){}^{2/3}+2 \sqrt [3]{3} e^{c_1} x}{6^{2/3} \sqrt [3]{\sqrt {81 x^6-12 e^{3 c_1} x^3}-9 x^3}}\right \},\left \{y(x)\to \frac {i \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+i\right ) \left (\sqrt {81 x^6-12 e^{3 c_1} x^3}-9 x^3\right ){}^{2/3}-2 \left (\sqrt {3}+3 i\right ) e^{c_1} x}{2\ 2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {81 x^6-12 e^{3 c_1} x^3}-9 x^3}}\right \},\left \{y(x)\to \frac {\sqrt [3]{2} \sqrt [6]{3} \left (-1-i \sqrt {3}\right ) \left (\sqrt {81 x^6-12 e^{3 c_1} x^3}-9 x^3\right ){}^{2/3}-2 \left (\sqrt {3}-3 i\right ) e^{c_1} x}{2\ 2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {81 x^6-12 e^{3 c_1} x^3}-9 x^3}}\right \}\right \}\]

Maple
cpu = 0.025 (sec), leaf count = 49

\[ \left \{ -\ln \left ({\frac {{x}^{2}-xy \relax (x ) + \left (y \relax (x ) \right ) ^{2}}{{x}^{2}}} \right ) -\ln \left ({\frac {x+y \relax (x ) }{x}} \right ) +\ln \left ({\frac {y \relax (x ) }{x}} \right ) -\ln \relax (x ) -{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[x*(x^3 - 2*y[x]^3)*y'[x] == y[x]*(2*x^3 - y[x]^3),y[x],x]

Mathematica raw output

{{y[x] -> (2*3^(1/3)*E^C[1]*x + 2^(1/3)*(-9*x^3 + Sqrt[-12*E^(3*C[1])*x^3 + 81*x
^6])^(2/3))/(6^(2/3)*(-9*x^3 + Sqrt[-12*E^(3*C[1])*x^3 + 81*x^6])^(1/3))}, {y[x]
 -> (-2*(3*I + Sqrt[3])*E^C[1]*x + I*2^(1/3)*3^(1/6)*(I + Sqrt[3])*(-9*x^3 + Sqr
t[-12*E^(3*C[1])*x^3 + 81*x^6])^(2/3))/(2*2^(2/3)*3^(5/6)*(-9*x^3 + Sqrt[-12*E^(
3*C[1])*x^3 + 81*x^6])^(1/3))}, {y[x] -> (-2*(-3*I + Sqrt[3])*E^C[1]*x + 2^(1/3)
*3^(1/6)*(-1 - I*Sqrt[3])*(-9*x^3 + Sqrt[-12*E^(3*C[1])*x^3 + 81*x^6])^(2/3))/(2
*2^(2/3)*3^(5/6)*(-9*x^3 + Sqrt[-12*E^(3*C[1])*x^3 + 81*x^6])^(1/3))}}

Maple raw input

dsolve(x*(x^3-2*y(x)^3)*diff(y(x),x) = (2*x^3-y(x)^3)*y(x), y(x),'implicit')

Maple raw output

-ln((x^2-x*y(x)+y(x)^2)/x^2)-ln((x+y(x))/x)+ln(y(x)/x)-ln(x)-_C1 = 0